### 20.3.1 Acoustic medium

Products: ABAQUS/Standard  ABAQUS/Explicit  ABAQUS/CAE

### Overview

An acoustic medium:

• is used to model sound propagation problems;

• can be used in a purely acoustic analysis or in a coupled acoustic-structural analysis such as the calculation of shock waves in a fluid or noise levels in a vibration problem;

• is an elastic medium (usually a fluid) in which stress is purely hydrostatic (no shear stress) and pressure is proportional to volumetric strain;

• is specified as part of a material definition;

• must appear in conjunction with a density definition (see Density, Section 16.2.1);

• can include fluid cavitation in ABAQUS/Explicit when the absolute pressure drops to a limit value;

• can be defined as a function of temperature and/or field variables;

• can include dissipative effects;

• can model small pressure changes (small amplitude excitation);

• can model waves in the presence of steady underlying flow of the medium; and

• is active only during dynamic analysis procedures (Dynamic analysis procedures: overview, Section 6.3.1).

### Defining an acoustic medium

The equilibrium equation for small motions of a compressible, inviscid fluid flowing through a resisting matrix material is taken to be

where p is the dynamic pressure in the fluid (the pressure in excess of any initial static pressure), is the spatial position of the fluid particle, is the fluid particle velocity, is the fluid particle acceleration, is the density of the fluid, and is the “volumetric drag” (force per unit volume per velocity) caused by the fluid flowing through the matrix material. The d'Alembert term has been written without convection on the assumption that there is no steady flow of the fluid, which is usually considered to be sufficiently accurate for steady fluid velocities up to Mach 0.1.

The constitutive behavior of the fluid is assumed to be inviscid and compressible, so that the bulk modulus of an acoustic medium relates the dynamic pressure in the medium to the volumetric strain by

where is the volumetric strain. Both the bulk modulus and the density of an acoustic medium must be defined.

The bulk modulus can be defined as a function of temperature and field variables but does not vary in value during an implicit dynamic analysis using the subspace projection method (Implicit dynamic analysis using direct integration, Section 6.3.2) or a direct-solution steady-state dynamic analysis (Direct-solution steady-state dynamic analysis, Section 6.3.4); for these procedures the value of the bulk modulus at the beginning of the step is used.

 Input File Usage: Use both of the following options to define an acoustic medium: ```*ACOUSTIC MEDIUM, BULK MODULUS *DENSITY```

 ABAQUS/CAE Usage: Property module: material editor: OtherAcoustic Medium: Bulk ModulusGeneralDensity

### Volumetric drag

Dissipation of energy (and attenuation of acoustic waves) may occur in an acoustic medium due to the fact that it is embedded in a matrix material that resists the flow of the medium. Such dissipation effects are commonly characterized in the frequency domain by a complex, frequency-dependent density, impedance, or propagation constant. In ABAQUS the flow resistance is modeled by a “volumetric drag coefficient” (force per unit volume per velocity). Conversion between these various approaches to modeling dissipation in the acoustic medium is discussed below.

If the acoustic medium is used in a direct-integration dynamic procedure (including ABAQUS/Explicit), the volumetric drag coefficient, , is assumed to be independent of frequency and the first value entered for the current temperature and/or field variable is used. can be entered as a function of frequency—, where f is the frequency in cycles per time (usually Hz)—in addition to temperature and/or field variables only when the acoustic medium is used in a steady-state dynamics procedure.

In all procedures except direct steady-state dynamics the gradient of is assumed to be small.

 Input File Usage: `*ACOUSTIC MEDIUM, VOLUMETRIC DRAG`

 ABAQUS/CAE Usage: Property module: material editor: OtherAcoustic Medium: Volumetric Drag: Include volumetric drag

#### Conversion between complex density and volumetric drag

The behavior of an acoustic medium is sometimes defined by its bulk modulus and a frequency-dependent “complex density,” . For use in ABAQUS/Standard this complex density must be converted to density, , and a volumetric drag coefficient, . Since the material density, , cannot be made a function of frequency in ABAQUS, only the frequency dependence of the imaginary part of the complex density can be modeled in a single ABAQUS/Standard analysis run. The conversion is

where and is the circular frequency, . Thus, and .

#### Conversion between complex impedance and volumetric drag

The complex impedance, Z, is related to the complex density by

so that

and

#### Conversion between a propagation constant and volumetric drag

The complex propagation constant for the medium, b, is related to the complex density by

Therefore,

and

### Fluid cavitation

In general, fluids cannot withstand any significant tensile stress and are likely to undergo large volume expansion when the absolute pressure is close to or less than zero. ABAQUS/Explicit allows modeling of this phenomenon through a cavitation pressure limit for the acoustic medium. When the fluid absolute pressure (sum of the dynamic and initial static pressures) reduces to this limit, the fluid undergoes free volume expansion (i.e., cavitation), without a further drop in the pressure. If this limit is not defined, the fluid is assumed not to undergo cavitation even under a tensile, negative absolute pressure, condition.

The constitutive behavior for an acoustic medium capable of undergoing cavitation can be stated as

where a pseudo-pressure , a measure of the volumetric strain, is defined as

is the fluid cavitation limit and is the initial acoustic static pressure. A total wave formulation is used for a nonlinear acoustic medium undergoing cavitation. This formulation is very similar to the scattered wave formulation except that the pseudo-pressure, defined as the product of the bulk modulus and the compressive volumetric strain, plays the role of the material state variable instead of the acoustic dynamic pressure and the acoustic dynamic pressure is readily available from this pseudo-pressure subject to the cavitation condition.

 Input File Usage: `*ACOUSTIC MEDIUM, CAVITATION LIMIT`

 ABAQUS/CAE Usage: Fluid cavitation is not supported in ABAQUS/CAE.

#### Defining the wave formulation

In the presence of cavitation in ABAQUS/Explicit the fluid mechanical behavior is nonlinear. Hence, for an acoustic problem with incident wave loading and possible cavitation in the fluid, the scattered wave formulation, which provides a solution for only a scattered wave dynamic acoustic pressure, may not be appropriate. For these cases the total wave formulation, which solves for the total dynamic acoustic pressure, should be selected. See Acoustic loads, Section 27.4.5, for details.

 Input File Usage: `*ACOUSTIC WAVE FORMULATION, TYPE=TOTAL WAVE `

 ABAQUS/CAE Usage: Any module: ModelEdit Attributesmodel_name. Toggle on Specify acoustic wave formulation: Total wave

#### Defining the initial acoustic static pressure

Cavitation occurs when the absolute pressure reaches the cavitation limit value. ABAQUS/Explicit allows for an initial linearly varying hydrostatic pressure in the fluid medium (see Defining initial acoustic static pressure” in “Initial conditions, Section 27.2.1). You can specify pressure values at two locations and a node set of the acoustic medium nodes. ABAQUS/Explicit interpolates from these data to initialize the static pressure at all the nodes in the specified node set. If the pressure at only one location is specified, the hydrostatic pressure in the fluid is assumed to be uniform. The acoustic static pressure is used only for determining the cavitation status of the acoustic element nodes and does not apply any static loads to the acoustic or structural mesh at their common wetted interface.

 Input File Usage: `*INITIAL CONDITIONS, TYPE=ACOUSTIC STATIC PRESSURE `

 ABAQUS/CAE Usage: Initial acoustic pressures are not supported in ABAQUS/CAE.

### Defining a steady flow field

Acoustic finite elements can be used to simulate time-harmonic wave propagation and natural frequency analysis in the presence of a steady mean flow of the medium. For example, air may move at a speed large enough to affect the propagation speed of waves in the direction of flow and against it. These effects are modeled in ABAQUS/Standard by specifying an acoustic flow velocity during the linear perturbation analysis step definition; you do not need to alter the acoustic material properties. See Acoustic, shock, and coupled acoustic-structural analysis, Section 6.9.1, for details.

### Elements

An acoustic material definition can be used only with the acoustic elements in ABAQUS (see Choosing the appropriate element for an analysis type, Section 21.1.3).

In ABAQUS/Standard second-order acoustic elements are more accurate than first-order elements. Use at least six nodes per wavelength in the acoustic medium to obtain accurate results.

### Output

Nodal output variable POR (pressure magnitude) is available for an acoustic medium in ABAQUS (in ABAQUS/CAE this output variable is called PAC). When the scattered wave formulation is used with incident wave loading in ABAQUS/Explicit, output variable POR represents only the scattered pressure response of the model and does not include the incident wave loading itself. When the total wave formulation is used, output variable POR represents the total dynamic acoustic pressure, which includes contributions from both incident and scattered waves as well as the dynamic effects of fluid cavitation. For either formulation output variable POR does not include the acoustic static pressure, which is used only to evaluate the cavitation status in the acoustic medium.

In addition, in ABAQUS/Standard nodal output variable PPOR (the pressure phase) is available for an acoustic medium. In ABAQUS/Explicit nodal output variable PABS (the absolute pressure, equal to the sum of POR and the acoustic static pressure) is available for an acoustic medium.